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Arens irregularity of the trace class convolution algebra
Author(s) -
Hu Zhiguo,
Neufang Matthias,
Ruan ZhongJin
Publication year - 2013
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bds093
Subject(s) - mathematics , banach algebra , locally compact group , dual polyhedron , trace (psycholinguistics) , convolution (computer science) , class (philosophy) , pure mathematics , algebra over a field , unitary state , locally compact space , trace class , banach space , hilbert space , linguistics , philosophy , machine learning , artificial intelligence , artificial neural network , computer science , law , political science
For any locally compact quantum group , the space T ( L 2 ()) of trace class operators on L 2 () is a Banach algebra with the convolution induced by the right fundamental unitary of . We study certain Arens irregularity properties of this convolution algebra. It is shown in particular that T ( L 2 ()) is right strongly Arens irregular in the sense of Dales and Lau (The second duals of Beurling algebras, Mem. Amer. Math. Soc. 177 (2005)) if and only if is finite. This generalizes a result by the second author on locally compact groups. We obtain a natural class of Banach algebras for which Arens regularity and strong Arens irregularity are, surprisingly, equivalent. We also give a precise description of the right topological centre of T ( L 2 ())** for all infinite discrete quantum groups with L 1 () strongly Arens irregular.